LINEAR NEUMANN VERTEX CONDITION
ISAAC WAHBI Version: July 9, 2018
Abstract. The purpose of this article is to study quasi linear parabolic partial differ-ential equations of second order, on a bounded junction, satisfying a nonlinear and non dynamical Neumann boundary condition at the junction point. We prove the existence and the uniqueness of a classical solution.
1. Introduction
In this paper, we study non degenerate quasi linear parabolic partial differential equa-tions on a junction, satisfying a non linear Neumann boundary condition at the junction point x = 0: ∂tui(t, x) − σi(x, ∂xui(t, x))∂x,x2 ui(t, x) + Hi(x, ui(t, x), ∂xui(t, x)) = 0,
for all x > 0, and for all i ∈ {1 . . . I}, F (u(t, 0), ∂xu(t, 0)) = 0.
(1)
The well-known Kirchhoff law corresponds to the case where F is linear in ∂xu and
independent of u.
Originally introduced by Nikol’skii [13] and Lumer [11, 12], the concept of ramified spaces and the analysis of partial differential equation on these spaces have attracted a lot of attention in the last 30 years. As explained in [13], the main motivations are applications in physics, chemistry, and biology (for instance small transverse vibrations in a grid of strings, vibration of a grid of beams, drainage system, electrical equation with Kirchhoff law, wave equation, heat equation,...). Linear diffusions of the form (1), with a Kirchhoff law, are also naturally associated with stochastic processes living on graphs. These processes were introduced in the seminal papers [3] and [4]. Another motivation for studying (1) is the analysis of associated stochastic optimal control problems with
a control at the junction. The result of this paper will allow us in a future work to characterize the value function of such problems.
There has been several works on linear and quasilinear parabolic equations of the form (1). For linear equations, von Below [15] shows that, under natural smoothness and compatibility conditions, linear boundary value problems on a network with a linear Kirchhoff condition and an additional Dirichlet boundary condition at the vertex point, are well-posed. The proof consists mainly in showing that the initial boundary value problem on a junction is equivalent to a well-posed initial boundary value problem for a parabolic system, where the boundary conditions are such that the classical results on linear parabolic equations [7] can be applied. The same author investigates in [16] the strong maximum principle for semi linear parabolic operators with Kirchhoff condition, while in [17] he studies the classical global solvability for a class of semilinear parabolic equations on ramified networks, where a dynamical node condition is prescribed: Namely the Neumann condition at the junction point x = 0 in (1), is replaced by the dynamic one
∂tu(t, 0) + F (t, u(t, 0), ∂xu(t, 0)) = 0.
In this way the application of classical estimates for domains established in [7] becomes possible. The author then establish the classical solvability in the class C1+α,2+α, with
the aid of the Leray-Schauder-principle and the maximum principle of [16]. Let us note that this kind of proof fails for equation (1) because in this case one cannot expect an uniform bound for the term |∂tu(t, 0)| (the proof of Lemma 3.1 of [7] VI.3 fails). Still
in the linear setting, another approach, yielding similar existence results, was developed by Fijavz, Mugnolo and Sikolya in [2]: the idea is to use semi-group theory as well as variational methods to understand how the spectrum of the operator is related to the structure of the network.
Equations of the form (1) can also be analyzed in terms of viscosity solutions. The first results on viscosity solutions for Hamilton-Jacobi equations on networks have been obtained by Schieborn in [14] for the Eikonal equations and later discussed in many contributions on first order problems [1, 6, 8], elliptic equations [9] and second order problems with vanishing diffusion at the vertex [10].
In contrast second order Hamilton-Jacobi equations with a non vanishing viscosity at the boundary have seldom been studied in the literature and our aim is to show the well-posedness of classical solutions for (1) in suitable H¨oder spaces: see Theorem 2.2 for the existence and Theorem 2.4 for the comparison, and thus the uniqueness. Our main assumptions are that the equation is uniformly parabolic with smooth coefficients and that the term F = F (u, p) at the junction is either decreasing with respect to u or increasing with respect to p.
The main idea of the proof is to use a time discretization, exploiting at each step the solvability in C2+α of the elliptic problem
−σi(x, ∂xui(x))∂x,x2 ui(x) + Hi(x, ui(x), ∂xui(x)) = 0, F (u(0), ∂xu(0)) = 0. (2)
The paper is organized as follows. In section 2, we introduce the notations and state our main results. In Section 3, we review the mains results of existence and uniqueness of the elliptic problem (2). Finally Section 4, is dedicated to the proof of our main results.
2. main results
In this section we state our main result Theorem 2.2, on the solvability of the parabolic problem with Neumann boundary condition at the vertex, on a bounded junction
∂tui(t, x) − σi(x, ∂xui(t, x))∂x,x2 ui(t, x)+ Hi(x, ui(t, x), ∂xui(t, x)) = 0, if (t, x) ∈ (0, T ) × (0, ai), F (u(t, 0), ∂xu(t, 0)) = 0, if t ∈ [0, T ), ∀i ∈ {1 . . . I}, ui(t, ai) = φi(t), if t ∈ [0, T ], ∀i ∈ {1 . . . I}, ui(0, x) = gi(x), if x ∈ [0, ai]. (3)
There will be two typical assumptions for F = F (u, p): either F is decreasing with respect to u or F is increasing with respect to p (Kirchhoff conditions).
2.1. Notations and preliminary results. Let us start by introducing the main nota-tion used in this paper as well as an interpolanota-tion result.
Let I ∈ N∗ be the number of edges, and a = (a
edge.
The bounded junction is defined by
Ja = n X = (x, i), x ∈ [0, a
i] and i ∈ {1, . . . , I}
o ,
where all the points (0, i), i = 1, . . . , I, are identified to the vertex denoted by 0. We can then write Ja = I [ i=1 Jai i , with Jai i := [0, ai] × {i}, Jiai∩ J aj
j = {0}. For T > 0, the time-space domain J a T is defined by Ja T = [0, T ] × J a. The interior of Ja
T set minus the junction point 0 is denoted by ◦ Ja T, and is defined by ◦ Ja T = (0, T ) × [I i=1 ◦ Jai i .
For the functionnal spaces that will be used in the sequel, we use here the notations of Chapter 1.1 of [7]. For the convenience of the reader, we recall these notations in Appendix A.
In addition we introduce the parabolic H¨older space on the junctionC2l,l(Ja
T), k.kC2 ,ll (Ja T)
and the space C
l 2,l b (
◦
Ja
T), defined by (where l > 0, see Annexe A for more details)
C2l,l(JTa) := n f : Ja T → R, (t, (x, i)) 7→ fi(t, x), ∀(i, j) ∈ {1 . . . I}2, ∀t ∈ (0, T ), fi(t, 0) = fj(t, 0) = f (t, 0), ∀i ∈ {1 . . . I}, (t, x) 7→ fi(t, x) ∈ C l 2,l([0, T ] × [0, ai]) o , C l 2,l b ( ◦ Ja T) := n f : Ja T → R, (t, (x, i)) 7→ fi(t, x), ∀i ∈ {1 . . . I}, (t, x) 7→ fi(t, x) ∈ C l 2,l b ((0, T ) × (0, ai)) o , with kuk C2 ,ll (Ja T) = X 1≤i≤I kuik C2 ,ll ([0,T ]×[0,ai]).
We will use the same notations, when the domain does not depend on time, namely T = 0, ΩT = Ω, removing the dependence on the time variable.
We continue with the definition of a nondecreasing maps F : RI → R.
Let (x = (x1, . . . xI), y = (y1. . . yI)) ∈ R2I, we say that
x ≤ y, if ∀i ∈ {1 . . . I}, xi ≤ yi,
and
x < y, if x ≤ y, and there exists j ∈ {1 . . . I}, xj < yj.
We say that F ∈ C(RI, R) is nondecreasing if
∀(x, y) ∈ RI, if x ≤ y, then F (x) ≤ F (y),
increasing if
∀(x, y) ∈ RI, if x < y, then F (x) < F (y).
Next we recall an interpolation inequality, which will be useful in the sequel.
Lemma 2.1. Suppose that u ∈ C0,1([0, T ] × [0, R]) satisfies an H¨older condition in t in
[0, T ] × [0, R], with exponent α ∈ (0, 1], constant ν1, and has derivative ∂xu, which for any
t ∈ [0, T ] are H¨older continuous in the variable x, with exponent γ ∈ (0, 1], and constant ν2. Then the derivative ∂xu satisfies in [0, T ] × [0, R], an H¨older condition in t, with
exponent αγ
1+γ, and constant depending only on ν1, ν2, γ. More precisely
∀(t, s) ∈ [0, T ]2, |t − s| ≤ 1, ∀x ∈ [0, R], |∂xu(t, x) − ∂xu(s, x)| ≤ 2ν2 ν1 γν2 1+γγ + 2ν1 γν2 ν1 −1+γ1 |t − s|1+γαγ .
This is a special case of Lemma II.3.1, in [7], (see also [13]). The main difference is that we are able to get global H¨older regularity in [0, T ] × [0, R] for ∂xu in its first variable.
Proof. Let (t, s) ∈ [0, T ]2, with |t − s| ≤ 1, and x ∈ [0, R]. Suppose first that x ∈ [0,R 2].
Let y ∈ [0, R], with y 6= x, we write
∂xu(t, x) − ∂xu(s, x) =
1 y − x
Z y
x
(∂xu(t, x) − ∂xu(t, z)) + (∂xu(t, z) − ∂xu(s, z)) + (∂xu(s, z) − ∂xu(s, x)) dz.
Using the H¨older condition in time satisfied by u, we have 1 y − x Z y x (∂xu(t, z) − ∂xu(s, z))dz ≤ 2ν1|t − s|α |y − x| .
On the other hand, using the H¨older regularity of ∂xu in space satisfied, we have
1 y − x Z y x
(∂xu(t, x) − ∂xu(t, z)) + (∂xu(s, z) − ∂xu(s, x))dz
≤ 2ν2|y − x| γ. It follows |∂xu(t, x) − ∂xu(s, x)| ≤ 2ν2|y − x|γ + 2ν1|t − s|α |y − x| . Assuming that |t − s| ≤(3R 2 ) 1+γ γν2 ν1 α1
∧ 1, minimizing in y ∈ [0, R], for y > x, the right side of the last equation, we get that the infimum is reached for
y∗ = x + ν1|t − s|α γν2 1+γ1 , and then |∂xu(t, x) − ∂xu(s, x)| ≤ C(ν1, ν2, γ)|t − s| αγ 1+γ,
where the constant C(ν1, ν2, γ), depends only on the data (ν1, ν2, γ), and is given by
C(ν1, ν2, γ) = 2ν2 ν1 γν2 1+γγ + 2ν1 γν2 ν1 −1+γ1 . For the cases y < x, and x ∈ [R
2, R], we argue similarly, which completes the proof.
2.2. Assumptions and main results. We state in this subsection the central Theo-rem of this note, namely the solvability and uniqueness of (1) in the class Cα2,1+α(Ja
T) ∩ C1+ α 2,2+α b ( ◦ Ja
Let us state the assumptions we will work on.
Assumption (P) We introduce the following data
F ∈ C0(R × RI, R) g ∈ C1(Ja) ∩ C2 b( ◦ Ja) ,
and for each i ∈ {1 . . . I}
σi ∈ C1([0, ai] × R, R) Hi ∈ C1([0, ai] × R2, R) φi ∈ C1([0, T ], R) .
We suppose furthermore that the data satisfy (i) Assumption on F
a) F is decreasing with respect to its first variable, b) F is nondecreasing with respect to its second variable, c) ∃(b, B) ∈ R × RI, F (b, B) = 0,
or satisfies the Kirchhoff condition
a) F is nonincreasing with respect to its first variable, b) F is increasing with respect to its second variable, c) ∃(b, B) ∈ R × RI, F (b, B) = 0.
We suppose moreover that there exist a parameter m ∈ R, m ≥ 2 such that we have (ii) The (uniform) ellipticity condition on the (σi)i∈{1...I}: there exists ν, ν, strictly positive
constants such that
∀i ∈ {1 . . . I}, ∀(x, p) ∈ [0, ai] × R,
ν(1 + |p|)m−2 ≤ σi(x, p) ≤ ν(1 + |p|)m−2.
(iii) The growth of the (Hi)i∈{1...I} with respect to p exceed the growth of the σi with
function such that
∀i ∈ {1 . . . I}, ∀(x, u, p) ∈ [0, ai] × R2, |Hi(x, u, p)| ≤ µ(|u|)(1 + |p|)m.
(iv) We impose the following restrictions on the growth with respect to p of the derivatives for the coefficients (σi, Hi)i∈{1...I}, which are for all i ∈ {1 . . . I},
a) |∂pσi|[0,ai]×R2(1 + |p|) 2+ |∂ pHi|[0,ai]×R2 ≤ γ(|u|)(1 + |p|) m−1, b) |∂xσi|[0,ai]×R2(1 + |p|) 2+ |∂ xHi|[0,ai]×R2 ≤ ε(|u|) + P (|u|, |p|)(1 + |p|)m+1, c) ∀(x, u, p) ∈ [0, ai] × R3, −CH ≤ ∂uHi(x, u, p) ≤ ε(|u|) + P (|u|, |p|)(1 + |p|)m,
where γ and ε are continuous non negative increasing functions. P is a continuous func-tion, increasing with respect to its first variable, and tends to 0 for p → +∞, uniformly with respect to its first variable, from [0, u1] with u1 ∈ R, and CH > 0 is real strictly
positive number. We assume that (γ, ε, P, CH) are independent of i ∈ {1 . . . I}.
(v) A compatibility conditions for g and (φi){1...I}
F (g(0), ∂xg(0)) = 0 ; ∀i ∈ {1 . . . I}, gi(ai) = φi(0).
Theorem 2.2.Assume (P). Then system (3) is uniquely solvable in the class Cα2,1+α(JTa)∩
C1+α2,2+α
b (
◦
Ja
T). There exist constants (M1, M2, M3), depending only the data introduced in
assumption (P), M1 = M1 maxi∈{1...I} n supx∈(0,ai)| − σi(x, ∂xgi(x))∂x2gi(x) + Hi(x, gi(x), ∂xgi(x))| + |∂tφi|(0,T ) o
, maxi∈{1...I}|gi|(0,ai), CH
, M2 = M2 ν, ν, µ(M1), γ(M1), ε(M1), sup|p|≥0P (M1, |p|), |∂xgi|(0,ai), M1 , M3 = M3 M1, ν(1 + |p|)m−2, µ(|u|)(1 + |p|)m, |u| ≤ M1, |p| ≤ M2 , such that ||u||C(Ja
Moreover, there exists a constant M(α) depending on α, M1, M2, M3 such that ||u||Cα 2 ,1+α(Ja T)≤ M(α).
We continue this Section by giving the definitions of super and sub solution, and stating a comparison Theorem for our problem.
Definition 2.3. We say that u ∈ C0,1(Ja
T) ∩ C1,2( ◦
Ja
T), is a super solution (resp. sub
solution) of ∂tui(t, x) − σi(x, ∂xui(t, x))∂x,x2 ui(t, x)+ Hi(x, ui(t, x), ∂xui(t, x)) = 0, if (t, x) ∈ (0, T ) × (0, ai), F (u(t, 0), ∂xu(t, 0)) = 0, if t ∈ (0, T ), (4) if ∂tui(t, x) − σi(x, ∂xui(t, x))∂x,x2 ui(t, x)+ Hi(x, ui(t, x), ∂xui(t, x)) ≥ 0, (resp. ≤ 0), ∀(t, x) ∈ (0, T ) × (0, ai),
F (u(t, 0), ∂xu(t, 0)) ≤ 0, (resp. ≥ 0), ∀t ∈ (0, T )
Theorem 2.4. Parabolic comparison. Assume (P). Let u ∈ C0,1(Ja T) ∩ C 1,2 b ( ◦ Ja T) (resp. v ∈ C0,1(J a T) ∩ C 1,2 b ( ◦ Ja T)) a super solution
(resp. a sub solution) of (4), satisfying for all i ∈ {1 . . . I}, ui(t, ai) ≥ vi(t, ai), for all
t ∈ [0, T ], and ui(0, x) ≥ vi(0, x), for all x ∈ [0, ai].
Then for each (t, (x, i)) ∈ Ja
T : ui(t, x) ≥ vi(t, x).
Proof. We start by showing that for each 0 ≤ s < T , for all (t, (x, i)) ∈ Ja
s, ui(t, x) ≥
vi(t, x).
Let λ > 0. Suppose that λ > C1+ C2, where the expression of the constants (C1, C2) are
given in the sequel (see (5), and (6)). We argue by contradiction assuming that sup (t,(x,i))∈Ja s exp(−λt + x)vi(t, x) − ui(t, x) > 0.
Using the boundary conditions satisfied by u and v, the supremum above is reached at a point (t0, (x0, j0)) ∈ (0, s] × J , with 0 ≤ x0 < aj0.
Suppose first that x0 > 0, the optimality conditions imply that exp(−λt0+ x0) − λ(vj0(t0, x0) − uj0(t0, x0)) + ∂tvj0(t0, x0) − ∂tuj0(t0, x0) ≥ 0, exp(−λt0+ x0)) vj0(t0, x0) − uj0(t0, x0) + ∂xvj0(t0, x0) − ∂xuj0(t0, x0) = 0, exp(−λt0+ x0) vj0(t0, x0) − uj0(t0, x0) + 2 ∂xvj0(t0, x0) − ∂xuj0(t0, x0) + ∂2 x,xvj0(t0, x0) − ∂ 2 x,xuj0(t0, x0) = exp(−λt0+ x0) −vj0(t0, x0) − uj0(t0, x0) + ∂2 x,xvj0(t0, x0) − ∂ 2 x,xuj0(t0, x0) ≤ 0. Using assumptions (P) (iv) a), (iv) c) and the optimality conditions above we have
Hj0(x0, ui(t0, x0), ∂xuj0(t0, x0)) − Hj0(x0, vj0(t0, x0), ∂xvj0(t0, x0)) ≤ vj0(t0, x0) − uj0(t0, x0) CH + γ(|∂xvj0(t0, x0)|) (1 + |∂xuj0(t0, x0))| ∨ |∂xvj0(t0, x0))|) m−1 ≤ C1 vj0(t0, x0) − uj0(t0, x0) , where C1 := maxi∈{1...I} n sup(t,x)∈[0,T ]×[0,ai]n CH + γ(|∂xvi(t, x)| 1 + |∂xui(t, x))| ∨|∂xvi(t, x))| m−1 o o . (5)
On the other hand we have using assumption (P) (ii), (iv) a), (iv) c), and the optimality conditions σj0(x0, ∂xvj0(t0, x0))∂ 2 x,xvj0(t0, x0) − σj0(x0, ∂xuj0(t0, x0))∂ 2 x,xuj0(t0, x0) ≤ vj0(t0, x0) − uj0(t0, x0) ν(1 + |∂xvj0(t0, x0)|) m−2 + ∂ 2 x,xuj0(t0, x0) + γ(|∂xuj0(t0, x0)|)(1 + |∂xuj0(t0, x0))| ∨ |∂xvj0(t0, x0))|) m−1 ≤ C2 vj0(t0, x0) − uj0(t0, x0) , where C2 := maxi∈{1...I} n sup(t,x)∈[0,T ]×[0,ai]n ν(1 + |∂xvi(t, x)|)m−2+ ∂ 2 x,xui(t, x) + γ(|∂xui(t, x)|)(1 + |∂xui(t, x))| + |∂xvi(t, x))|)m−1 o o . (6)
Using now the fact that v is a sub-solution while u is a super-solution, we get 0 ≤ ∂tuj0(t0, x0) − σj0(x0, ∂xuj0(t0, x0))∂ 2 x,xuj0(t0, x0) + Hj0(x0, ui(t0, x0), ∂xuj0(t0, x0)) −∂tvj0(t0, x0) + σj0(x0, ∂xvj0(t0, x0))∂ 2 x,xvj0(t0, x0) − Hj0(x0, vj0(t0, x0), ∂xvj0(t0, x0)) ≤ −(λ − (C1+ C2))(vj0(t0, x0) − uj0(t0, x0)) < 0,
which is a contradiction. Therefore the supremum is reached at (t0, 0), with t0 ∈ (0, s].
We apply a first order Taylor expansion in space, in the neighborhood of the junction point 0. Since for all (i, j) ∈ {1 . . . I}, ui(t0, 0) = uj(t0, 0) = u(t0, 0), and vi(t0, 0) =
vj(t0, 0) = v(t0, 0), we get from ∀(i, j) ∈ {1, . . . I}2, ∀h ∈ (0, min i∈{1...I}ai] vj(t0, 0) − uj(t0, 0) ≥ exp(h) vi(t0, h) − ui(t0, h) , that ∀(i, j) ∈ {1, . . . I}2, ∀h ∈ (0, min i∈{1...I}ai] vj(t0, 0) − uj(t0, 0) ≥ vi(t0, 0) − ui(t0, 0) + h vi(t0, 0) − ui(t0, 0) + ∂xvi(t0, 0) − ∂xui(t0, 0) + hεi(h), where
∀i ∈ {1, . . . I}, limh→0εi(h) = 0.
We get then ∀i ∈ {1, . . . I}, ∂xvi(t0, 0) ≤ ∂xui(t0, 0) − vi(t0, 0) − ui(t0, 0) < ∂xui(t0, 0).
Using the growth assumptions on F (assumption (P)(i)), and the fact that v is a sub-solution while u is a super-sub-solution, we get
and then a contradiction.
We deduce then for all 0 ≤ s < T , for all (t, (x, i)) ∈ [0, s] × Ja,
exp(−λt + x)vi(t, x) − ui(t, x)
≤ 0.
Using the continuity of u and v, we deduce finally that for all (t, (x, i)) ∈ [0, T ] × Ja,
vi(t, x) ≤ ui(t, x).
3. The elliptic problem
As explained in the introduction, the construction of a solution for our parabolic prob-lem (3) relies on a time discretization and on the solvability of the associated elliptic problem. We review in this section the well-posedness of the elliptic problem (2), which is formulated for regular maps (x, i) 7→ ui(x), continuous at the junction point, namely
each i 6= j ∈ {1 . . . I}, ui(0) = uj(0) = u(0), that follows at each edge
−σi(x, ∂xui(x))∂x,x2 ui(x) + Hi(x, ui(x), ∂xui(x)) = 0,
and ui satisfy the following non linear Neumann boundary condition at the vertex
F (u(0), ∂xu(0)) = 0, where ∂xu(0) = (∂xu1(0), . . . , ∂xuI(0)).
We introduce the following data for i ∈ {1 . . . I} F ∈ C(R × RI, R), σi ∈ C1([0, ai] × R, R) Hi ∈ C1([0, ai] × R2, R) φi ∈ R ,
satisfying the following assumptions
(i) Assumption on F
a) F is decreasing with respect to its first variable, b) F is nondecreasing with respect to its second variable, c) ∃(b, B) ∈ R × RI, such that : F (b, B) = 0,
or F satisfy the Kirchhoff condition
a) F is nonincreasing with respect to its first variable, b) F is increasing with respect to its second variable, c) ∃(b, B) ∈ R × RI, such that : F (b, B) = 0.
(ii) The ellipticity condition on the σi
∃c > 0, ∀i ∈ {1 . . . I}, ∀(x, p) ∈ [0, ai] × R, σi(x, p) ≥ c.
(iii) For the Hamiltonians Hi, we suppose
∃CH > 0, ∀i ∈ {1 . . . I}, ∀(x, u, v, p) ∈ (0, ai) × R3,
if u ≤ v, CH(u − v) ≤ Hi(x, u, p) − Hi(x, v, p).
For each i ∈ {1 . . . I}, we define the following differential operators (δi, δi)i∈{1...I} acting
on C1([0, a
i] × R2, R), for f = f (x, u, p) by
δi := ∂u+
1
p∂x; δi := p∂p.
(iv) We impose the following restrictions on the growth with respect to p for the coefficients (σi, Hi)i∈{1...I} = (σi(x, p), Hi(x, u, p))i∈{1...I}, which are for all i ∈ {1 . . . I}
δiσi = o(σi),
δiσi = O(σi),
Hi = O(σip2),
δiHi ≤ o(σip2),
where the limits behind are understood as p → +∞, uniformly in x, for bounded u. The main result of this section is the following Theorem, for the solvability and uniqueness of the elliptic problem at the junction, with non linear Neumann condition at the junction point.
Theorem 3.1. Assume (E). The following elliptic problem at the junction, with Neumann boundary condition at the vertex
−σi(x, ∂xui(x))∂2x,xui(x) + Hi(x, ui(x), ∂xui(x)) = 0, if x ∈ (0, ai), F (u(0), ∂xu(0)) = 0, ∀i ∈ {1 . . . I}, ui(ai) = φi, (7)
is uniquely solvable in the class C2+α(Ja).
Theorem 3.1 is stated without proof in [9]. For the convenience of the reader, we sketch its proof in the Appendix.
The uniqueness of the solution of (7), is a consequence of the elliptic comparison Theorem for smooth solutions, for the Neumann problem, stated in this Section, and whose proof uses the same arguments of the proof of the parabolic comparison Theorem 2.4. We complete this section by recalling the definition of super and sub solution for the elliptic problem (7), and the corresponding elliptic comparison Theorem.
Definition 3.2. Let u ∈ C2(Ja). We say that u is a super solution (resp. sub solution)
of −σi(x, ∂xfi(x))∂x,x2 fi(x) + Hi(x, fi(x), ∂xfi(x)) = 0, if x ∈ (0, ai), F (f (0), ∂xf (0)) = 0, (8) if −σi(x, ∂xui(x))∂x,x2 ui(x) + Hi(x, ui(x), ∂xui(x)) ≥ 0, (resp. ≤ 0), if x ∈ (0, ai),
F (u(0), ∂xu(0)) ≤ 0, (resp. ≥ 0).
Theorem 3.3. Elliptic comparison Theorem, see for instance Theorem 2.1 of [9].
of (8), satisfying for all i ∈ {1 . . . I}, ui(ai) ≥ vi(ai). Then for each (x, i) ∈ Ja :
ui(x) ≥ vi(x).
4. The parabolic problem
In this Section, we prove Theorem 2.2. The construction of the solution is based on the results obtained in Section 3 for the elliptic problem, and is done by considering a sequence un∈ C2(Ja), solving on a time grid an elliptic scheme defined by induction. We
will prove that the solution un converges to the required solution.
4.1. Estimates on the discretized scheme. Let n ∈ N∗, we consider the following
time grid, (tn k =
kT
n)0≤k≤n of [0, T ], and the following sequence (uk)0≤k≤n of C
2+α(Ja),
defined recursively by
for k = 0, u0 = g,
and for 1 ≤ k ≤ n, uk is the unique solution of the following elliptic problem
n(ui,k(x) − ui,k−1(x)) − σi(x, ∂xui,k(x))∂x,x2 ui,k(x))+
Hi(x, ui,k(x), ∂xui,k(x)) = 0, if x ∈ (0, ai),
F (uk(0), ∂xuk(0)) = 0,
∀i ∈ {1 . . . I}, ui,k(ai) = φi(tnk).
(9)
The solvability of the elliptic scheme (9) can be proved by induction, using the same arguments as for Theorem 3.1. The next step consists in obtaining uniform estimates of (uk)0≤k≤n. We start first by getting uniform bounds for n|uk− uk−1|(0,ai) using the
comparison Theorem 3.3.
Lemma 4.1. Assume (P). There exists a constant C > 0, independent of n, depending only the data C = Cmaxi∈{1...I}
n supx∈(0,ai)|−σi(x, ∂xgi(x))∂ 2 xgi(x)+Hi(x, gi(x), ∂xgi(x))|+ |∂tφi|(0,T ) o , CH , such that sup n≥0 max k∈{1...n} i∈{1...I}max n
n|ui,k− ui,k−1|(0,ai)
o
and then sup n≥0 max k∈{0...n} i∈{1...I}max n |ui,k|(0,ai) o ≤ C + max i∈{1...I} n |gi|(0,ai) o .
Proof. Let n > ⌊CH⌋, where CH is defined in assumption (P) (iv) c). Let k ∈ {1 . . . n},
we define the following sequence M0 = maxi∈{1...I} n supx∈(0,ai)| − σi(x, ∂xgi(x))∂ 2 xgi(x) + Hi(x, gi(x), ∂xgi(x))| + |∂tφi|(0,T ) o , Mk,n = n n − CH Mk−1,n, k ∈ {1 . . . n}.
We claim that for each k ∈ {1 . . . n} max
i∈{1...I}
n
n|ui,k− ui,k−1|(0,ai)
o
≤ Mk,n.
We give a proof by induction. For this, if k = 1, let us show that the map h defined on the junction by h := Ja → R (x, i) 7→ M1,n n + gi(x),
is a super solution of (9), for k = 1. For this we will use the Elliptic Comparison Theorem 3.3.
Using the compatibility conditions satisfied by g, namely assumption (P) (v), and the assumptions of growth on F , assumption (P) (i), we get for the boundary conditions
F (h(0), ∂xh(0)) = F ( M1,n n + g(0), ∂xg(0)) ≤ F (g(0), ∂xg(0)) = 0, h(ai) = M1,n n + gi(ai) ≥ M0,n n + gi(ai) ≥ φi(t n 1).
For all i ∈ {1 . . . I}, and x ∈ (0, ai), we get using assumption (P) (iii)
n(hi(x) − gi(x)) − σi(x, ∂xhi(x))∂x2hi(x) + Hi(x, hi(x), ∂xhi(x)) = M1,n− σi(x, ∂xgi(x))∂x2gi(x) + Hi(x, M1,n n + gi(x), ∂xgi(x)) ≥ M1,n− σi(x, ∂xgi(x))∂x2gi(x) + Hi(x, gi(x), ∂xgi(x)) − M1,nCH n ≥ 0.
It follows from the comparison Theorem 3.3, that for all i ∈ {1 . . . I}, and x ∈ [0, ai]
u1,i(x) ≤
M1,n
n + gi(x). Using the same arguments, we show that
h := Ja → R (x, i) 7→ −M1,n n + gi(x),
is a sub solution of (9) for k = 1, and we then get max i∈{1...I} n sup x∈(0,a) n|u1,i(x) − gi(x)| o ≤ M1,n.
Let 2 ≤ k ≤ n, suppose that the assumption of induction holds true. Let us show that the following map
h := Ja→ R (x, i) 7→ Mk,n n + ui,k−1(x),
is a super solution of (9). For the boundary conditions, using assumption (P) (i), we get F (h(0), ∂xh(0)) = F ( Mk,n n + uk−1(0), ∂xuk−1(0)) ≤ F (uk−1(0), ∂xuk−1(0)) ≤ 0, h(ai) = Mk,n n + ui,k−1(ai) ≥ M0,n n + ui,k−1(ai) ≥ φi(t n k).
For all i ∈ {1 . . . I}, and x ∈ (0, ai)
n(hi(x) − ui,k−1(x)) − σi(x, ∂xh(x))∂x2h(x) + Hi(x, h(x), ∂xh(x)) =
Mk,n− σi(x, ∂xui,k−1(x))∂x2ui,k−1(x) + Hi(x,
Mk,n
n + ui,k−1(x), ∂xuk−1(x)) ≥ Mk,n − σi(x, ∂xui,k−1(x))∂2xui,k−1(x) + Hi(x, ui,k−1(x), ∂xuk−1(x)) −
CHMk,n
n .
Since we have for all x ∈ (0, ai)
using the induction assumption we get
n(hi(x) − ui,k−1(x)) − σi(x, ∂xh(x))∂x2h(x) + Hi(x, ∂xh(x), ∂xh(x)) ≥
Mk,n− n(ui,k−1(x) − ui,k−2(x)) −
CHMk,n
n ≥ Mk,n
n − CH
n − Mk−1,n ≥ 0.
It follows from the comparison Theorem 3.3, that for all (x, i) ∈ Ja
ui,k(x) ≤
Mk,n
n + ui,k−1(x). Using the same arguments, we show that
h := Ja→ R (x, i) 7→ −Mk,n n + ui,k−1(x),
is a sub solution of (9), and we get max
i∈{1...I}
n
n|ui,k(x) − ui,k−1(x)|(0,ai)
o
≤ Mk,n.
We obtain finally using that for all k ∈ {1 . . . n} Mk,n ≤ Mn,n, Mk,n = n n − CH k M0, and Mn,n n→+∞
−−−−→ M := exp(CH) maxi∈{1...I}
n supx∈(0,ai)| − σi(x, ∂xgi(x))∂x2gi(x) + Hi(x, gi(x), ∂xgi(x))| + |∂tφi|(0,T ) o , that
supn≥0 maxk∈{1...n} maxi∈{1...I}
n
n|ui,k− ui,k−1|(0,ai)
o
≤ C, supn≥0 maxk∈{0...n} maxi∈{1...I}
n |ui,k|(0,ai) o ≤ C + maxi∈{1...I} n |gi|(0,ai) o .
That completes the proof.
The next step consists in obtaining uniform estimates for |∂xuk|(0,ai), in terms of n|uk−
(iv). More precisely, we use similar arguments as for the proof of Theorem 14.1 of [5], using a classical argument of upper and lower barrier functions at the boundary. The assumption of growth (P) (ii) and (iii) are used in a key way to get an uniform bound on the gradient at the boundary. Finally to conclude, we appeal to a gradient maximum principle, using the growth assumption (P) (iv), adapting Theorem 15.2 of [5] to our elliptic scheme.
Lemma 4.2. Assume (P). There exists a constant C > 0, independent of n, depending only the data
ν, ν, µ(|u|), γ(|u|), ε(|u|), sup|p|≥0P (|u|, |p|), |∂xgi|(0,ai),
|u| ≤ supn≥0maxk∈{0...n}maxi∈{1...I}
n
|ui,k|(0,ai)
o , supn≥0maxk∈{1...n}maxi∈{1...I}
n
n|ui,k− ui,k−1|(0,ai)
o , such that sup n≥0 max k∈{0...n} i∈{1...I}max n |∂xui,k|(0,ai) o ≤ C.
Proof. Step 1 : We claim that, for each k ∈ {1 . . . n}, maxi∈{1...I}
n
|∂xui,k|∂(0,ai)
o is bounded by the data, uniformly in n.
It follows from Lemma 4.1, that there exists M > 0 such that sup
n≥0 k∈{0...n}max i∈{1...I}max
n
|ui,k|(0,ai)+ n|ui,k− ui,k−1|(0,ai)
o
≤ M.
We fix i ∈ {1 . . . I}. We apply a barrier method consisting in building two functions wi,k+, wi,k− satisfying in a neighborhood of 0, for example [0, κ], with κ ≤ ai
Qi(x, w+i,k(x), ∂xw+i,k(x), ∂x2w+i,k(x)) ≥ 0, ∀x ∈ [0, κ], wi,k+(0) = ui,k(0), wi,k+(κ) ≥ M,
Qi(x, wi,k−(x), ∂xw−i,k(x), ∂x2w−i,k(x)) ≤ 0, ∀x ∈ [0, κ], w −
i,k(0) = ui,k(0), w −
i,k(κ) ≤ −M,
where we recall that for each (x, u, p, S) ∈ [0, ai] × R3
For n > ⌊CH⌋, where CH is defined in assumption P (iv) c), it follows then from the
comparison principle that
wi,k−(x) ≤ ui,k(x) ≤ w+i,k(x), ∀x ∈ [0, κ],
and then
∂xwi,k−(0) ≤ ∂xui,k(0) ≤ ∂xw+i,k(0).
We look for wi,k+ defined on [0, κ] of the form w+ i,0 = gi(x) w+ i,k : x 7→ ui,k(0) + 1 β ln(1 + θx),
where the constants (β, θ, κ) will be chosen in the sequel independent of k. Remark first that for all x ∈ [0, κ], ∂2
xwi,k+(x) = −β∂xw +
i,k(x)2, and w +
i,k(0) = ui,k(0). Let
we choose (θ, κ), such that
∀k ∈ {1 . . . n}, 0 < κ ≤ min
i∈{1...I}ai, w +
i,k(κ) ≥ M, ∂xw+i,k(κ) ≥ β. (10)
We choose for instance
θ = β2exp(2βM) + 1 mini∈{1...I}ai exp(2βM) κ = 1 θ exp(2βM) − 1. (11)
The constant β will be chosen in order to get
β ≥ sup k∈{1...n} sup x∈[0,κ] µ(wi,k+(x))(1 + ∂xwi,k+(x))m+ M ν(1 + ∂xwi,k+(x))m−2∂xwi,k+(x)2 , (12)
where (µ(.), ν, m) are defined in assumption (P) (ii) and (iii). Since we have ∀x ∈ [0, κ], w+i,k(x) ≤ wi,k+(κ) = 2M,
β ≤ ∂xwi,k+(κ) ≤ ∂xw+i,k(x) ≤ ∂xw+i,k(0).
We can then choose β large enough to get (12), for instance
β ≥ µ(2M) ν 1 + 1 β2 + M νβ2.
It is easy to show by induction that w+i,k is lower barrier of ui,kin the neighborhood [0, κ].
More precisely, since w+i,0 = ui,0, and for all k ∈ {1 . . . n}
wi,k+(0) = ui,k(0), w+i,k(κ) ≥ ui,k(κ),
w+i,k(x) = w+i,k−1(x) + ui,k(0) − ui,k−1(0) ≥ wi,k−1+ (x) −
M n ,
we get using the assumption of induction, assumption (P) (ii) and (iii), and (12) that for all x ∈ (0, κ)
n(wi,k+(x) − ui,k−1(x)) − σi(x, ∂xw+i,k(x))∂x,xwi,k+(x) + Hi(x, w+i,k(x), ∂xw+i,k(x)) ≥
−M + βσi(x, ∂xw+i,k(x))∂xwi,k+(x)2+ Hi(x, wi,k+(x), ∂xwi,k+(x)) ≥
−M + βν(1 + ∂xwi,k+(x))m−2∂xw+i,k(x)2+ µ(w+i,k(x))(1 + ∂xw+i,k(x))m≥ 0.
We obtain therefore sup
n≥0
max
k∈{0...n} i∈{1...I}max ∂xui,k(0) ≤
θ
β ∨ ∂xgi(0). With the same arguments we can show that
wi,0− = gi(x)
wi,k− : x 7→ ui,k(0) −
1
β ln(1 + θx),
is a lower barrier in the neighborhood of 0. Using the same method, we can show that ∂xui,k(ai) is uniformly bounded by the same upper bounds, which completes the proof of
Step 1.
Step 2 : For the convenience of the reader, we do not detail all the computations of this Step, since they can be found in the proof of Theorem 15.2 of [5]. It follows from Lemma 4.1 that there exists M > 0 such that
supn≥0 maxk∈{0...n} maxi∈{1...I}
n
|ui,k|(0,ai)
o
≤ M. We set furthermore
Let u be a solution of the elliptic equation, for x ∈ (0, ai)
σi(x, ∂xu(x))∂x,xu(x) − Hi,kn (x, u(x), ∂xu(x)) = 0,
and assume that |u|(0,ai) ≤ M. The main key of the proof will be in the use of the following
equalities δiHi,kn (x, u, p) = δiHi(x, u, p) + n(p − ∂xui,k−1(x)) p , δiH n i,k(x, u, p) = δiHi(x, u, p), (13)
where we recall that the operators δi and ¯δi are defined in assumption (E) (iii). We follow
the proof of Theorem 15.2 in [5]. We set u = ψ(u), where ψ ∈ C3[m, M ], is increasing
and m = φ(−M), M = φ(M). In the sequel, we will set v = ∂xu2 and v = ∂xu2. To
simplify the notations, we will omit the variables (x, u(x), ∂xu(x)) in the functions σi and
Hn
i,k, and the variable u for ψ. We assume first that the solution u ∈ C3([−M, M]), and
we follow exactly all the computations that lead to equation of (15.25) of [5] to get the following inequality
σi∂x,xv + Bi∂xv + Gni,k ≥ 0, (14)
where Bi and Gni,k have the same expression in (15.26) of [5] with (σi = σi∗, ci = 0). We
choose (r = 0, s = 0), since we will see in the sequel (15), that condition (15.32) of [5] holds under assumption assumption (P). We have more precisely
Bi = ψ′∂pσi∂x,xu − ∂pHi+ ω∂p(σip2), Gn i,k= ω′ ψ′ + κiω 2+ β iω + θi,kn , ω = ψ ′′ ψ′2 ∈ C 1([m, M]), κi = 1 σip2 δi(σip2) + p2 4σi |(δi+ 1)σi|2 , βi = 1 σip2 δi(σip2) − δiHi+ p2 2σi ((δi+ 1)σi)(δiσi) , θn i,k = 1 σip2 p2 4σi |δiσi|2− δiHi,kn = θi− 1 σip2 n(p − ∂xu i,k−1(x)) p , θi = 1 σip2 p2 4σi |δiσi|2− δiHi .
We set in the sequel Gi =
∂xω
∂xψ
+ κiω2+ βiω + θi, in order to get Gni,k = Gi−
1 σip2 n(p − ∂xui,k−1(x)) p . More precisely, we see from (13) that all the coefficients (Bi, κi, βi, θi) can be chosen
independent of n and ui,k−1. The main argument then to get a bound of ∂xu is to apply
a maximum principle for v in (14), and this will be done as soon as we ensure Gn
i,k ≤ 0, for |∂xu| ≥ Lnk.
On the other hand, using assumption (P) (ii) (iii) and (iv), it is easy to check that there exists a constants (a, b, c), depending only on the data
ν, ν, µ(M), γ(M), ε(M), sup
|p|≥0
P (M, |p|), such that
supx∈[0,ai],|u|≤M lim sup|p|→+∞ κi(x, u, p) ≤ a,
supx∈[0,ai],|u|≤M lim sup|p|→+∞ βi(x, u, p) ≤ b,
supx∈[0,ai],|u|≤M lim sup|p|→+∞ θi(x, u, p) ≤ c,
where a = 1 ν(γ(M) + ν) + 1 2+ γ(M)2 ν2 , b = ε(M) + sup|p|≥0P (M, |p|) + γ(M) ν + (ε(M) + sup|p|≥0P (M, |p|))(ν + γ(M)) ν2 , c = (ε(M) + sup|p|≥0P (M, |p|)) 2 4ν2 + 2(ε(M) + sup|p|≥0P (M, |p|)) ν .
As it has been on the proof of Theorem 15.2 of [5], we choose then L = L(a, b, c), and ψ(·) = ψ(a, b, c)(·) such that we have
Gi ≤ 0, if |∂xu(x)| ≥ L(a, b, c).
We see then from the expression of θn
i,k that we get
Gn
Therefore applying the maximum principle to v in (14), and from the relation u = ψ(u), v = ∂xu2 we get finally
|∂xu|(0,ai)≤ max
max ψ′(a, b, c)(·)
min ψ′(a, b, c)(·), |∂xu|∂(0,ai), L(a, b, c), |∂xui,k−1|(0,ai)
. This upper bound still holds if u ∈ C2([0, a
i]), (cf. (15.30) and (15.31) of the proof of
Theorem 15.2 in [5]). Finally applying the upper bound above to the solution uk, we get
by induction that
supn≥0 maxk∈{0...n} maxi∈{1...I}
n
|∂xui,k|(0,ai)
o
≤ maxmax ψ
′(a, b, c)(·)
min ψ′(a, b, c)(·), |∂xui,k|∂(0,ai), L(a, b, c), |∂xgi|(0,ai)
.
This completes the proof.
The following Proposition follows from Lemmas 4.1 and 4.2, assumption (P) (ii) (iii), and from the relation
∀x ∈ [0, ai], |∂x,x2 ui,k(x))| ≤
|n(ui,k(x) − ui,k−1(x))| + |Hi(x, ui,k(x), ∂xui,k(x))|
σi(x, ∂xui,k(x))
≤ |n(ui,k(x) − ui,k−1(x))| + µ(|ui,k(x)|)(1 + |∂xui,k(x)|
m)
ν(1 + |∂xui,k(x)|m−2)
.
Proposition 4.3. Assume (P). There exist constants (M1, M2, M3), depending only the
data introduced in assumption (P)
M1 = M1 maxi∈{1...I} n supx∈(0,ai)| − σi(x, ∂xgi(x))∂x2gi(x) + Hi(x, gi(x), ∂xgi(x))| + |∂tφi|(0,T ) o
, maxi∈{1...I}|gi|(0,ai), CH
, M2 = M2 ν, ν, µ(M1), γ(M1), ε(M1), sup|p|≥0P (M1, |p|), |∂xgi|(0,ai), M1 , M3 = M3 M1, ν(1 + |p|)m−2, µ(|u|)(1 + |p|)m, |u| ≤ M1, |p| ≤ M2 ,
such that sup n≥0 max k∈{0...n} i∈{1...I}max n |ui,k|(0,ai) o ≤ M1, sup n≥0 max k∈{0...n} i∈{1...I}max n |∂xui,k|(0,ai) o ≤ M2, sup n≥0 max k∈{1...n} i∈{1...I}max n
|n(ui,k− ui,k−1)|(0,ai)
o ≤ M1, sup n≥0 max k∈{0...n} i∈{1...I}max n |∂x,xui,k|(0,ai) o ≤ M3.
Unfortunately, we are unable to give an upper bound of the modulus of continuity of ∂2
x,xui,k in Cα([0, a]) independent of n. However, we are able to formulate in the weak
sense a limit solution. From the regularity of the coefficients, using some tools introduced in Section 1, Lemma 2.1, we get interior regularity, and a smooth limit solution.
4.2. Proof of Theorem 2.2.
Proof. The uniqueness is a result of the comparison Theorem 2.4. To simplify the
nota-tions, we set for each i ∈ {1 . . . I}, and for each (x, q, u, p, S) ∈ [0, ai] × R4
Qi(x, u, q, p, S) = q − σi(x, p)S + Hi(x, u, p).
Let n ≥ 0. Consider the subdivision (tn k =
kT
n )0≤k≤n of [0, T ], and (uk)0≤k≤n the solution
of (9).
From estimates of Proposition 4.3, there exists a constant M > 0 independent of n, such that
supn≥0 maxk∈{1...n} maxi∈{1...I}
n
|ui,k|(0,ai) + |n(ui,k− ui,k−1)|(0,ai) +
|∂xui,k|(0,ai) + |∂x,xui,k|(0,ai)
o
≤ M. (15)
We define the following sequence (vn)n≥0in C0,2(JTa), piecewise differentiable with respect
to its first variable by
∀i ∈ {1 . . . I}, vi,0(0, x) = gi(x) if x ∈ [0, ai],
We deduce then from (15), that there exists a constant M1 independent of n, depending
only on the data of the system, such that for all i ∈ {1 . . . I} |vi,n|α[0,T ]×[0,ai] + |∂xvi,n|
α
x,[0,T ]×[0,ai] ≤ M1.
Using Lemma 2.1, we deduce that there exists a constant M2(α) > 0, independent of n,
such that for all i ∈ {1 . . . I}, we have the following global H¨older condition |∂xvi,n| α 2 t,[0,T ]×[0,ai] + |∂xvi,n| α x,[0,T ]×[0,ai] ≤ M2(α).
We deduce then from Ascoli’s Theorem, that up to a sub sequence n, (vi,n)n≥0 converge
in C0,1([0, T ] × [0, a
i]) to vi, and then vi ∈ C α
2,1+α([0, T ] × [0, ai]).
Since vn satisfies the following continuity condition at the junction point
∀(i, j) ∈ {1 . . . I}2, ∀n ≥ 0, ∀t ∈ [0, T ], vi,n(t, 0) = vj,n(t, 0) = vn(t, 0),
we deduce then v ∈ Cα2,1+α(Ja T).
We now focus on the regularity of v inJ◦a
T, and we will prove that v ∈ C1+ α 2,2+α(
◦
Ja T), and
satisfies on each edge
Qi(x, vi(t, x), ∂tvi(t, x), ∂xvi(t, x), ∂x,x2 vi(t, x)) = 0, if (t, x) ∈ (0, T ) × (0, ai).
Using once again (15), there exists a constant M3 independent of n, such that for each
i ∈ {1 . . . I}
k∂tvi,nkL2((0,T )×(0,ai)) ≤ M3, k∂ 2
x,xvi,nkL2((0,T )×(0,ai)) ≤ M3.
Hence we get up to a sub sequence, that
∂tvi,n ⇀ ∂tvi, ∂x,x2 vi,n ⇀ ∂x,x2 vi,
weakly in L2((0, T ) × (0, ai)).
the variable ∂t and ∂x,x2 , allows us to get for each i ∈ {1 . . . I}, up to a subsequence np Z T 0 Z ai 0
Qi(x, vi,np(t, x), ∂tvi,np(t, x), ∂xvi,np(t, x), ∂ 2 x,xvi,np(t, x)) ψ(t, x)dxdt p→+∞ −−−−→ Z T 0 Z ai 0 Qi(x, vi(t, x), ∂tvi(t, x), ∂xvi(t, x), ∂2x,xvi(t, x)) ψ(t, x)dxdt, ∀ψ ∈ C∞ c ((0, T ) × (0, ai)).
We now prove that for any ψ ∈ C∞
c ((0, T ) × (0, ai)) Z T 0 Z ai 0
Qi(x, vi,np(t, x), ∂tvi,np(t, x), ∂xvi,np(t, x), ∂ 2
x,xvi,np(t, x)))
ψ(t, x)dxdt −−−−→ 0.p→+∞ Using that (uk)0≤k≤n is the solution of (9), we get for any ψ ∈ Cc∞((0, T ) × (0, ai))
Z T
0
Z ai 0
Qi(x, vi,n(t, x), ∂tvi,n(t, x), ∂xvi,n(t, x), ∂x,x2 vi,n(t, x))
ψ(t, x)dxdt = n−1 X k=0 Z tnk+1 tn k Z ai 0
σi(x, ∂xui,k+1(x))∂x,x2 ui,k+1(x) − σi(x, ∂xvi,n(t, x))∂x,x2 vi,n(t, x)
+Hi(x, vi,n(t, x), ∂xvi,n(t, x)) − Hi(x, ui,k+1(x), ∂xui,k+1(x))
ψ(t, x)dxdt. (16) Using assumption (P) more precisely the Lipschitz continuity of the Hamiltonians Hi, the
H¨older equicontinuity in time of (vi,n, ∂xvi,n), there exists a constant M4(α) independent
of n, such that for each i ∈ {1 . . . I}, for each (t, x) ∈ [tn
k, tnk+1] × [0, ai]
|Hi(x, ui,k+1(x), ∂xui,k+1(x)) − Hi(x, vi,n(t, x), ∂xvi,n(t, x))| ≤ M4(α)(t − tnk) α 2,
and therefore for any ψ ∈ C∞
c ((0, T ) × (0, ai)) n−1 X k=0 Z tnk+1 tn k Z ai 0
Hi(x, ui,k+1(x), ∂xui,k+1(x)) − Hi(x, vi,n(t, x), ∂xvi,n(t, x))
ψ(t, x)dxdt ≤ aiM4(α)|ψ|(0,T )×(0,ai)n −α 2 −−−−→ 0.n→+∞
For the last term in (16), we write for each i ∈ {1 . . . I}, for each (t, x) ∈ (tn k, t
n
k+1) × (0, ai)
σi(x, ∂xui,k+1(x))∂x,x2 ui,k+1(x) − σi(x, ∂xvi,n(t, x))∂x,x2 vi,n(t, x) =
σi(x, ∂xui,k+1(x)) − σi(x, ∂xvi,n(t, x)) ∂2 x,xui,k(x) + (17) σi(x, ∂xui,k+1(x)) − n(t − tnk)σi(x, ∂xvi,n(t, x)) ∂2 x,xui,k+1(x) − ∂x,x2 ui,k(x) . (18)
Using again the H¨older equicontinuity in time of (vi,n, ∂xvi,n) as well as the uniform bound
on |∂2
x,xui,k|[0,ai] (15), we can show that for (17), for any ψ ∈ C ∞ c ((0, T ) × (0, ai)), n−1 X k=0 Z tnk+1 tn k Z ai 0 σi(x, ∂xui,k+1(x)) − σi(x, ∂xvi,n(t, x)) ∂x,x2 ui,k(x)ψ(t, x)dxdt n→+∞ −−−−→ 0.
Finally, from assumptions (P), for all i ∈ {1 . . . I}, σi is differentiable with respect to all
its variable, integrating by part we get for (18) n−1 X k=0 Z tnk+1 tn k Z ai 0 σi(x, ∂xui,k+1(x)) − n(t − tnk)σi(x, ∂xvi,n(t, x)) ∂2 x,xui,k+1(x) − ∂2x,xui,k(x) ψ(t, x)dxdt = n−1 X k=0 Z tnk+1 tn k Z ai 0 ∂x σi(x, ∂xui,k+1(t, x))ψ(t, x) − n(t − tnk)∂x σi(x, ∂xvi,n(t, x))ψ(t, x) ∂xui,k+1(x) − ∂xui,k(x) dxdt n→+∞ −−−−→ 0. We conclude that for any ψ ∈ Cc∞((0, T ) × (0, ai))
Z T 0 Z ai 0 Qi(x, vi(t, x), ∂tvi(t, x), ∂xvi(t, x), ∂x,x2 vi(t, x))) ψ(t, x)dxdt = 0.
It is then possible to consider the last equation as a linear one, with coefficients ˜σi(t, x) =
σi(x, ∂xvi(t, x)), ˜Hi(t, x) = Hi(x, vi(t, x), ∂xvi(t, x)) belonging to the class C α
2,α((0, T ) ×
(0, ai)), and using Theorem III.12.2 of [7], we get finally that for all i ∈ {1 . . . I}, vi ∈
C1+α
2,2+α((0, T ) × (0, ai)), which means that v ∈ C1+ α 2,2+α(
◦
Ja T).
We deduce that vi satisfies on each edge
Qi(x, vi(t, x), ∂tvi(t, x), ∂xvi(t, x), ∂2x,xvi(t, x))) = 0, if (t, x) ∈ (0, T ) × (0, ai).
From the estimates (15), we know that ∂tvi,n and ∂x,x2 vi,n are uniformly bounded by n.
We deduce finally that v ∈ C1+α2,2+α
b (
◦
Ja T).
We conclude by proving that v satisfies the non linear Neumann boundary condition at the vertex. For this, let t ∈ (0, T ); we have up to a sub sequence np
F (vnp(t, 0), ∂xvnp(t, 0)) −−−−→
On the other hand, using that F (uk(0), ∂0uk(x)) = 0, we know from the continuity of F
(assumption (P)), the H¨older equicontinuity in time of t 7→ vn(t, 0), and t 7→ ∂xv(t, 0),
that there exists a constant M5(α) independent of n, such that if t ∈ [tnk, tnk+1)
|F (vn(t, 0), ∂xvn(t, 0))| = |F (vn(t, 0), ∂xvn(t, 0)) − F (uk(0), ∂xuk(0))| ≤
supn|F (u, x) − F (v, y)|, |u − v| + kx − ykRI ≤ M5(α)n− α 2
o n→+∞ −−−−→ 0.
Therefore, we conclude once more from the continuity of F (assumption (P)), the com-patibility condition (assumption (P) (v)), that for each t ∈ [0, T )
F (v(t, 0), ∂xv(t, 0)) = 0.
On the other hand, it is easy to get
∀i ∈ {1 . . . I}, ∀x ∈ [0, ai], vi(0, x) = gi(x), ∀t ∈ [0, T ], vi(t, ai) = φi(t).
Finally, the expression of the upper bounds of the solution given in Theorem 2.2, are a consequence of Proposition 4.3, and Lemma 2.1, which completes the proof.
4.3. On the existence for unbounded junction. We give in this subsection a result on the existence and the uniqueness of the solution for the parabolic problem (1), in a unbounded junction J defined for I ∈ N∗ edges by
J = nX = (x, i), x ∈ R+ and i ∈ {1, . . . , I} o . In the sequel, C0,1(J T) ∩ C1,2( ◦
JT) is the class of function with regularity C0,1([0, T ] ×
[0, +∞)) ∩ C1,2((0, T ) × (0, +∞)) on each edge, and L∞(J
T) is the set of measurable real
bounded maps defined on JT.
We introduce the following data F ∈ C0(R × RI, R) g ∈ C1 b(J ) ∩ Cb2( ◦ J ) ,
and for each i ∈ {1 . . . I} σi ∈ C1(R+× R, R) Hi ∈ C1(R+× R2, R) φi ∈ C1([0, T ], R) .
We suppose furthermore that the data satisfy the following assumption Assumption (P∞) (i) Assumption on F
a) F is decreasing with respect to its first variable, b) F is nondecreasing with respect to its second variable, c) ∃(b, B) ∈ R × RI, F (b, B) = 0,
or the Kirchhoff condition
a) F is nonincreasing with respect to its first variable, b) F is increasing with respect to its second variable, c) ∃(b, B) ∈ R × RI, F (b, B) = 0.
We suppose moreover that there exist a parameter m ∈ R, m ≥ 2 such that we have (ii) The (uniform) ellipticity condition on the (σi)i∈{1...I}: there exists ν, ν, strictly positive
constants such that
∀i ∈ {1 . . . I}, ∀(x, p) ∈ R+× R,
ν(1 + |p|)m−2 ≤ σi(x, p) ≤ ν(1 + |p|)m−2.
(iii) The growth of the (Hi)i∈{1...I} with respect to p exceed the growth of the σi with
respect to p by no more than two, namely there exists µ an increasing real continuous function such that
(iv) We impose the following restrictions on the growth with respect to p of the derivatives for the coefficients (σi, Hi)i∈{1...I}, which are for all i ∈ {1 . . . I},
a) |∂pσi|R+×R2(1 + |p|)2+ |∂pHi|R+×R2 ≤ γ(|u|)(1 + |p|)m−1, b) |∂xσi|R+×R2(1 + |p|)2+ |∂xHi|R+×R2 ≤ ε(|u|) + P (|u|, |p|)(1 + |p|)m+1, c) ∀(x, u, p) ∈ R+× R2, −CH ≤ ∂uHi(x, u, p) ≤ ε(|u|) + P (|u|, |p|)(1 + |p|)m,
where γ and ε are continuous non negative increasing functions. P is a continuous func-tion, increasing with respect to its first variable, and tends to 0 for p → +∞, uniformly with respect to its first variable, from [0, u1] with u1 ∈ R, and CH > 0 is real strictly
positive number. We assume that (γ, ε, P, CH) are independent of i ∈ {1 . . . I}.
(v) A compatibility conditions for g
F (g(0), ∂xg(0)) = 0.
We state here a comparison Theorem for the problem 1, in a unbounded junction. Theorem 4.4. Assume (P∞). Let u ∈ C0,1(JT)∩C1,2(
◦
JT)∩L∞(JT) (resp. v ∈ C0,1(JT)∩
C1,2(J◦
T) ∩ L∞(JT)) be a super solution (resp. a sub solution) of (4) (where ai = +∞),
satisfying for all i ∈ {1 . . . I} for all x ∈ [0, +∞), ui(0, x) ≥ vi(0, x). Then for each
(t, (x, i)) ∈ JT : ui(t, x) ≥ vi(t, x).
Proof. Let s ∈ [0, T ), K = (K . . . K) > (1, . . . 1) in RI, and λ = λ(K) > 0, that will be
chosen in the sequel. We argue as in the proof of Theorem 2.4, assuming sup (t,(x,i))∈JK s exp(−λt − (x − 1) 2 2 ) vi(t, x) − ui(t, x) > 0.
Using the boundary conditions satisfied by u and v, the above supremum is reached at a point (t0, (x0, j0)) ∈ (0, s] × J , with 0 ≤ x0 ≤ K.
If x0 ∈ [0, K), the optimality conditions are given for x0 6= 0 by −λ(vj0(t0, x0) − uj0(t0, x0)) + ∂tvj0(t0, x0) − ∂tuj0(t0, x0) ≥ 0, −(x0− 1) vj0(t0, x0) − uj0(t0, x0) + ∂xvj0(t0, x0) − ∂xuj0(t0, x0) = 0, vj0(t0, x0) − uj0(t0, x0) − 2(x0 − 1)2 vj0(t0, x0) − uj0(t0, x0) + ∂2 x,xvj0(t0, x0) − ∂ 2 x,xuj0(t0, x0) ≤ 0, and if x0 = 0, ∀i ∈ {1, . . . I}, ∂xvi(t0, 0) ≤ ∂xui(t0, 0) − vi(t0, 0) − ui(t0, 0) < ∂xui(t0, 0).
If x0 = 0, we obtain a contradiction exactly as in the proof of Theorem 2.4. On the other
hand if x0 ∈ (0, K), using assumptions (P) (iv) a), (iv) c) and the optimality conditions,
we can choose λ(K) of the form λ(K) = C(1 + K2), (see (5) and (6)), where C > 0 is a
constant independent of K, to get again a contradiction. We deduce that, if sup (t,(x,i))∈JK s exp(−λ(K)t − (x − 1) 2 2 ) vi(t, x) − ui(t, x) > 0,
then for all (t, (x, i)) ∈ [0, T ] × JK
exp(−λ(K)t − (x − 1) 2 2 ) vi(t, x) − ui(t, x) ≤ exp(−λ(K)t − (K − 1) 2 2 ) vi(t, K) − ui(t, K) . Hence for all (t, (x, i)) ∈ [0, T ] × JK
exp(−(x − 1) 2 2 ) vi(t, x) − ui(t, x) ≤ exp(−(K − 1) 2 2 ) vi(t, K) − ui(t, K) . On the other hand, if
sup (t,(x,i))∈JK s exp(−λ(K)t − (x − 1) 2 2 ) vi(t, x) − ui(t, x) ≤ 0,
then for all (t, (x, i)) ∈ [0, T ] × JK
exp(−λ(K)t − (x − 1) 2 2 ) vi(t, x) − ui(t, x) ≤ 0. So exp(−(x − 1) 2 2 ) vi(t, x) − ui(t, x) ≤ 0.
Finally we have, for all (t, (x, i)) ∈ [0, T ] × JK max0, exp(−(x − 1) 2 2 ) vi(t, x) − ui(t, x) ≤ exp(−(K − 1) 2 2 ) ||u||L∞(J T)+ ||v||L∞(JT) . Sending K → ∞ and using the boundedness of u and v, we deduce the inequality v ≤ u
in [0, T ] × J .
Theorem 4.5. Assume (P∞). The following parabolic problem with Neumann boundary
condition at the vertex
∂tui(t, x) − σi(x, ∂xui(t, x))∂x,x2 ui(t, x)+ Hi(x, ui(t, x), ∂xui(t, x)) = 0, if (t, x) ∈ (0, T ) × (0, +∞), F (u(t, 0), ∂xu(t, 0)) = 0, if t ∈ [0, T ), ∀i ∈ {1 . . . I}, ui(0, x) = gi(x), if x ∈ [0, +∞), (19)
is uniquely solvable in the class Cα2,1+α(JT)∩C1+ α 2,2+α(
◦
JT). There exist constants (M1, M2, M3),
depending only the data introduced in assumption (P∞)
M1 = M1 maxi∈{1...I} n supx∈(0,+∞)| − σi(x, ∂xgi(x))∂x2gi(x) + Hi(x, gi(x), ∂xgi(x))| o , maxi∈{1...I}|gi|(0,+∞), CH , M2 = M2 ν, ν, µ(M1), γ(M1), ε(M1), sup|p|≥0P (M1, |p|), |∂xgi|(0,+∞), M1 , M3 = M3 M1, ν(1 + |p|)m−2, µ(|u|)(1 + |p|)m, |u| ≤ M1, |p| ≤ M2 , such that
||u||C(JT) ≤ M1, ||∂xu||C(JT) ≤ M2, ||∂tu||C(JT) ≤ M1, ||∂x,xu||C(JT)≤ M3.
Moreover, there exists a constant M(α) depending on α, M1, M2, M3
such that for any
a ∈ (0, +∞)I
||u||Cα 2 ,1+α(Ja
Proof. Assume (P∞) and let a = (a, . . . , a) ∈ (0, +∞)I. Applying Theorem 2.2, we can
define ua∈ C0,1(Ja
T) ∩ C1,2( ◦
Ja
T) as the unique solution of
∂tui(t, x) − σi(x, ∂xui(t, x))∂x,x2 ui(t, x)+ Hi(x, ui(t, x), ∂xui(t, x)) = 0, if (t, x) ∈ (0, T ) × (0, a), F (u(t, 0), ∂xu(t, 0)) = 0, if t ∈ [0, T ),
∀i ∈ {1 . . . I}, ui(t, a) = gi(a), if t ∈ [0, T ],
∀i ∈ {1 . . . I}, ui(0, x) = gi(x), if x ∈ [0, a].
(20)
Using assumption (P∞) and Theorem 2.2, we get that there exists a constant C > 0
independent of a such that
supa≥0 ||ua||C1,2(Ja
T) ≤ C.
We are going to send a to +∞ in (20).
Following the same argument as for the proof of Theorem 2.2, we get that, up to a sub sequence, ua converges locally uniformly to some map u which solves (19). On the
other hand, uniqueness of u is a direct consequence of the comparison Theorem 4.4, since u ∈ L∞(J
T). Finally the expression of the upper bounds of the derivatives of u given in
Theorem 4.5, are a consequence of Theorem 2.2 and assumption (P∞).
Appendix A. Functionnal spaces
In this section, we recall several classical notations from [7]. Let l, T ∈ (0, +∞) and Ω be an open and bounded subset of Rnwith smooth boundary (n > 0). We set Ω
T = (0, T )×Ω,
and we introduce the following spaces : -if l ∈ 2N∗, Cl
2,l(ΩT), k · k
C2 ,ll (ΩT)
is the Banach space whose elements are continuous functions (t, x) 7→ u(t, x) in ΩT, together with all its derivatives of the form ∂tr∂xsu, with
2r + s < l. The norm k · k
C2 ,ll (ΩT) is defined for all u ∈ C l 2,l(ΩT) by kuk C2 ,ll (ΩT) = X 2r+s=j sup (t,x)∈ΩT |∂r t∂xsu(t, x)|. -if l /∈ N∗, Cl 2,l(ΩT), k.k C2l,l(ΩT)
functions (t, x) 7→ u(t, x) in ΩT, together with all its derivatives of the form ∂tr∂xsu, with
2r + s < l, and satisfying an H¨older condition with exponent l−2r−s
2 in their first variable,
and with exponent (l − ⌊l⌋) in their second variable, over all the connected components of ΩT whose radius is smaller than 1.
The norm k · k
C2 ,ll (ΩT)
is defined for all u ∈ C2l,l(ΩT) by
kuk C2l,l(ΩT) = |u| l ΩT + ⌊l⌋ X j=0 |u|jΩT, with ∀j ∈ {0, . . . , l}, |u|jΩT = X 2r+s=j sup (t,x)∈ΩT |∂r t∂xsu(t, x)|, |u|l ΩT = |u| l x,ΩT + |u| l 2 t,ΩT, |u|l x,ΩT = X 2r+s=⌊l⌋ |∂r t∂ s xu(t, x)| l−⌊l⌋ x,ΩT, |u|l t,ΩT = X 0<l−2r−s<2 |∂r t∂ s xu(t, x)| l−2r−s 2 t,ΩT , |u|α x,ΩT = sup t∈(0,T ) sup x,y∈Ω,x6=y,|x−y|≤1
|u(t, x) − u(t, y)|
|x − y|α , 0 < α < 1, |u|α t,ΩT = sup x∈Ω sup t,s∈(0,T ),t6=s,|t−s|≤1 |u(t, x) − u(s, x)| |t − s|α , 0 < α < 1.
- C2l,l(ΩT) is the set whose elements f belong to C l
2,l(OT) for any open set OT separated
from the boundary of ΩT by a strictly positive distance, namely
inf y∈∂ΩT,x∈OT ||x − y||Rn > 0. - C l 2,l b (ΩT) is the subset of C l
2,l(ΩT) consisting in maps u such that the derivatives of the
form ∂r
t∂sxu, (with 2r + s < l) are bounded, namely sup(t,x)∈ΩT|∂ r
t∂xsu(t, x)| < +∞.
We use the same notations when the domain does not depend on time, namely T = 0, ΩT = Ω, just removing the dependence on the time variable.
For R > 0, we denote by L2((0, T ) × (0, R)) the usual space of square integrable maps and
by C∞
c ((0, T ) × (0, R)) the set of infinite continuous differentiable functions on (0, T ) ×
Appendix B. The Elliptic problem
Proposition B.1. Let θ ∈ R, i ∈ {1, . . . , I} and assume (E) holds. Let uθ
i ∈ C2([0, ai]) be the solution to −σi(x, ∂xuθi(x))∂x,x2 uθi(x) + Hi(x, uθi(x), ∂xuθi(x)) = 0, if x ∈ (0, ai) uθ i(0) = uθ(0) = θ, uθ i(ai) = φi. (21)
Then the following map
Ψ := R→ C2([0, ai]) θ 7→ uθ i is continuous.
Proof. Let θn a sequence converging to θ. Using the Schauder estimates Theorem 6.6 of
[5], we get that there exists a constant M > 0 independent of n, depending only the data, such that for all α ∈ (0, 1)
kuθn
i kC2+α([0,a
i]) ≤ M.
From Ascoli’s Theorem, uθn
i converges up to a subsequence to v in C2([0, ai]) solution of
(21). By uniqueness of the solution of (21), uθn
i converges necessary to the solution uθi of
(21) in C2([0, a
i]), which completes the proof.
Proof of Theorem 3.1.
Proof. The uniqueness of (7) results from the elliptic comparison Theorem 3.3.
We turn to the solvalbility, and for this let θ ∈ R. We consider the elliptic Dirichlet problem at the junction
−σi(x, ∂xui(x))∂x,x2 ui(x) + Hi(x, ui(x), ∂xui(x)) = 0, if x ∈ (0, ai),
∀i ∈ {1 . . . I}, ui(0) = u(0) = θ,
ui(ai) = φi.
For all i ∈ {1 . . . I}, each elliptic problem is uniquely solvable on each edge in C2+α([0, a i]),
then (22) is uniquely solvable in the class C2+α(Ja), and we denote by uθ its solution.
We turn to the Neumann boundary condition at the vertex. Let us recall assumption (E)(i)
F is decreasing in its first variable, nondecreasing in its second variable, or F is nonincreasing in its first variable, increasing in its second variable, ∃(b, B) ∈ R × RI, such that : F (b, B) = 0. Fix now Ki = sup (x,u)∈(0,ai)×(−aiBi,aiBi) |Hi(x, u, Bi)|, θ ≥ |b| + max i∈{1...I} n |φi| + |aiBi| + Ki CH o ,
and let us show that f : x 7→ θ + Bix, is a super solution on each edge Jiai of (22).
We have the boundary conditions
f (0) = θ, f (ai) = θ + aiBi ≥ |φi| + |aiBi| + aiBi ≥ φi,
and using assumption (E) (iii), we have for all x ∈ (0, ai)
−σi(x, ∂xf (x))∂x,x2 f (x) + Hi(x, f (x), ∂xf (x)) = Hi(x, θ + Bix, Bi) ≥ Hi(x, Bix, Bi)
+ CHθ ≥ Hi(x, Bix, Bi) + Ki ≥ 0.
We then get that for each i ∈ {1 . . . I}, x ∈ [0, ai], uθi(x) ≤ θ+Bix, and a Taylor expansion
in the neighborhood of the junction point gives that for each i ∈ {1 . . . I}, ∂xuθi(0) ≤ Bi.
Since uθ(0) = θ ≥ b, we then get from assumption (E) (i)
F (uθ(0), ∂ xuθ(0)) ≤ F (b, B) ≤ 0. Similarly, fixing θ ≤ − |b| − min i∈{1...I} n − |φi| − |aiBi| − Ki CH o ,
the map f : x 7→ θ + xBi is a sub solution on each vertex Jiai of (22), then for each
i ∈ {1 . . . I}, ∂xuθi(0) ≥ Bi, which means
F (uθ(0), ∂
xuθ(0)) ≥ 0.
From Proposition B.1, we know that the real maps θ 7→ uθ(0) and θ 7→ ∂
xuθ(0) are
continuous. Using the continuity of F (assumption (E)), we get that θ 7→ F (uθ(0), ∂
xuθ(0))
is continuous, and therefore there exists θ∗ ∈ R such that
F (uθ∗(0), ∂xuθ ∗
(0)) = 0.
We remark that θ∗ is bounded by the data, namely θ∗ belongs to the following interval
h − |b| − max i∈{1...I} n |φi| + |aiBi| + sup(x,u)∈(0,ai)|Hi(x,Bix,Bi)| CH o , |b| + max i∈{1...I} n |φi| + |aiBi| + sup(x,u)∈(0,ai)|Hi(x,Bix,Bi)| CH o i .
This completes the proof. Finally, since the solution uθ∗
of (7) is unique, we get the uniqueness of θ∗.
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